Covers in free lattices
Authors:
Ralph Freese and J. B. Nation
Journal:
Trans. Amer. Math. Soc. 288 (1985), 142
MSC:
Primary 06B25
MathSciNet review:
773044
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Abstract: In this paper we study the covering relation in finitely generated free lattices. The basic result is an algorithm which, given an element , finds all the elements which cover or are covered by (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in are classified; again, with finitely many exceptions, they are all one, two or threeelement chains.
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 , Splitting lattices generate all lattices, Algebra Universalis 7 (1977), 163169. MR 0434897 (55:7861)
 [4]
 , Characterizations of lattices that are boundedhomomorphic images of sublattices of free lattices, Canad. J. Math. 31 (1979), 6978. MR 518707 (81h:06004)
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 A. Day and J. B. Nation, A note on finite sublattices of free lattices, Algebra Universalis 15 (1982), 9094. MR 663955 (83k:06011)
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 R. A. Dean, Coverings in free lattices, Bull. Amer. Math. Soc. 67 (1961), 548549. MR 0133264 (24:A3098)
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 R. Freese, Ideal lattices of lattices, Pacific J. Math. 57 (1975), 125133. MR 0371751 (51:7968)
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 , Some order theoretic questions about free lattices and free modular lattices, Proc. Banff Sympos. on Ordered Sets, Reidel, Dordrecht, 1982.
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 [10]
 G. Grätzer, General lattice theory, Academic Press, New York, 1978.
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 B. Jónsson and J. Kiefer, Finite sublattices of a free lattice, Canad. J. Math. 14 (1962), 487497. MR 0137667 (25:1117)
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 B. Jónsson and J. B. Nation, A report on sublattices of a free lattice, Colloq. Math. Soc. János Bolyai, Contributions to Universal Algebra (Szeged), Vol. 17, NorthHolland, Amsterdam, 1977, pp. 223257. MR 0472614 (57:12310)
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 R. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 143. MR 0313141 (47:1696)
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 J. B. Nation, Bounded finite lattices, Colloq. Math. Soc. János Bolyai, Contributions to Universal Algebra (Esztergom), Vol. 29, NorthHolland, Amsterdam, 1982, pp. 531533. MR 660892 (83f:06009)
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 , Finite sublattices of a free lattice, Trans. Amer. Math. Soc. 269 (1982), 311337. MR 637041 (83b:06008)
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 [18]
 , Free lattices. II, Ann. of Math. (2) 43 (1942), 104115. MR 0006143 (3:261d)
 [19]
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DOI:
http://dx.doi.org/10.1090/S00029947198507730444
PII:
S 00029947(1985)07730444
Article copyright:
© Copyright 1985
American Mathematical Society
